The temperature in Gavin\'s oven is a sinusoidal function of time. Gavin sets hi

Question

The temperature in Gavin's oven is a sinusoidal function of time. Gavin sets his oven so that it has a maximum temperature of 310 degree F and a minimum temperature of 230 degree, Once the temperature hits 310 degree, it takes 20 minutes before it is 310 degree again. Gavin's cake needs to be in the oven for 30 minutes at temperatures at or above 290 degree. He puts the cake into the oven when it is at 270 degree and rising. How long will Gavin need to leave the cake in the oven? Suppose T(t) = 24 sin(2 pi/24(t - 7)) + 63 is the temperature (in degrees Fahrenheit) at time t, where t is measured in hours after midnight on Sunday. You finish painting the exterior door to your house at 5 PM on Monday. The paint information states that 48 hours of 71 degree F drying time is required: i.e., you can only count time periods when the temperature is at least 71 degree F. When will the door be dry?

Explanation / Answer

General sine function format

A*sin(w*t)

what you know here is the frequency the and amplitude of the oven's temperature

we know the temperature fluctuates between 240-300, with a median at 270, 270 is 30 off of both of them, so we will consider 270 to be out "y=0" and 30 to be the amplitude

we also know that w = 2*pi*f
F = frequency, which we know that the oven goes through a 20 minute cycle so it has frequency of 1/20 cycles per minute

so f = 1/20

we now have an equation of

T = 30*sin(2*pi/20 * t)
we will add 270 to this so that out temperature match up and make it a little less confusing

T = 30*sin(2*pi/20 * t) + 270

so we need to know when this oven is going to reach 290 degrees, so set the equation to 290

290 = 30*sin(2*pi/20 * t) + 270

2/3 = sin(2pi*t/20)

do some rearranging to solve for t
t = 20*asin(2/3) / 2*pi
t = 2.218 minutes

so we know it is in the oven for 1.0817 minutes before it heats up to 290 degrees. This is the up part of the sin wave.

We also know that the wave will also be 270 degrees halfway through its cycle, which is 10 minutes in. since sine waves are symmetric, we know it will stop being 290 degrees 2.218 minutes before this point

so the leading and trailing 2.218 minutes can be removed so we get

10 - 2.218 *2 = 5.564 minutes. in the "up" part of the sine wave cycle
we know that the last 10 minutes of the cycle will all be below 270 degrees, so that is no good,

so basically we get a 5.564 minutes per cycle if we take this, so 30 / 5.564 = 5.3 cycles

from that we see that we need at least 5 cycles, not a full 5, so this means that he will take the cake out partway through.

we need to find that point

so if it went through a full 5 cycles we know it took 60 minutes at least

He must leave it in the oven for 5 full cycles;
during the 6th cycle he can remove it after
2.218 + (30 - 5*5.564)
=4.423 minutes.

Annswere is 64.423 minutes.

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